Date: 03 April 94
SUBJECT: Fermat again!
There has been a really amazing development today
on Fermat’s Last Theorem.
Noam Elkies has announced a counterexample,
so that Fermat’s Last Theorem is not true after all!
He spoke about this at the Institute today.
The solution to Fermat
that he constructs
involves an incredibly large
prime exponent
(larger than 10^20),
but it is constructive.
The main idea seems to be
a kind of Heegner point construction,
combined with a really ingenious descent
for passing from the modular curves
to the Fermat curve.
The really difficult part of the argument
seems to be
to show that the field of definition of the solution
(which, a priori,
is some ring class field
of an imaginary quadratic field)
actually descends to Q.
I wasn’t able to get all the details,
which were quite intricate . . .
So it seems that
the Taniyama-Shimura conjecture
is not true after all.
The experts think that it can still be salvaged,
by extending the concept of automorphic representation,
and introducing a notion of
"anomalous curves"
that would still give rise to a
"quasiautomorphic representation."
Henri Darmon
Princeton University
FERMAT’s Last Theorem
30 Nov. 2006
The terse format of FERMAT’s
formidible notation,
in the margin of
the page,
next to the formulation:
Xn
+ Yn = Zn ,
suggests that he considered the problem to be
easily understood.
He alliterated in Latin,
what they allude in Manhattan :
“ It is impossible for a cube
to be the sum of two cubes,
a fourth power to be the sum
of two fourth powers,
or in general
for any number that is a power
greater than the second
to be the sum
of two like powers.”
The goal of this article is to examine
WHY:
An
can not be equated
in Real Integers
with the Sum of
fewer than
n terms
of the
same exponent?
(FERMAT
states:
“the sum of two
like powers.”)
FERMAT’s Last
Theorem
states that Non-Zero
Integer solutions
do not exist for the equation:
An =
Bn + Cn,
when: n > 2.
The case of: n = one (1)
is a relation of entities having One Dimention:
A1 =
B1 + C1
71 =
31 +
41
This is trivial, except for the implication that the terms
are commensurable. All terms are One Dimention - lines.
Only one term is needed to the right of the equal sign.
The surplus of terms available to Equate the summation,
allow more solutions of the other terms
when any of the terms is fixed.
The case of:
n = two (2)
is a relation of entities having Two Dimentions:
Two brothers bet the captain
that his boat would go faster with their two overcoats,
than his sail.
The Area extent of each big windbag,
in two Dimentions is:
Bob (3 ft2)
and
Charles (4 ft2):
A2 =
B2 + C2
52 =
32 +
42
and
(x * 5)2 =
(x * 3)2 +
(x * 4)2
There are Non-Zero
Integer solutions for
n = 3,
when another term is added.
This equates the same quantity of terms
to the Sum, as the Exponent
of the terms:
A3 =
B3 + C3 +
D3
63 =
33 + 43 + 53
and
(x * 6)3 =
(x * 3)3 +
(x * 4)3 +
(x * 5)3
Simon Singh explains in his book: Fermat’s
Enigma, page 20,
Pythagoras first demonstrated—
his theorem is true for ALL
right-angled triangles.
“But how did Pythagoras know that his theorem is true
for every right-angled triangle? ”
Pythagoras was a pioneer, similar to
Pierre de FERMAT; because he developed mathematical proofs,
three hundred years
before Aristotle identified LOGIC.
Approximately the same period elapsed before Andrew WILES
developed the mathematical building blocks to “prove”
the “mathematical monster” (pages 29 -30)
that FERMAT created by increasing the exponent power
of Pythagoras’ Theorem.
Simon Singh explains on page 14:
“Pythagoras was also intrigued by the link between
numbers and nature. He realized that
natural phenomena are governed by laws,
and that these laws could be described by
mathematical equations.
One of the first links he discovered was the
fundamental relationship between the harmony of music
and the harmony of numbers.”
This is the goal of my analysis—
focus attention on the limited nature of existence,
and the quantitative relations between each of the particulars.
Why is FERMAT’s observation valid?
The relation of numbers means a relation of
things which exist. What do the numbers MEAN?
“I’ve got your Number,”
Pythagoras could have said,
to assert that he understood the essence of your character.
We understand that qualities of honesty or deceit,
determination or velleity, explain the activities
which a person chooses to perform.
Can we understand that the “Number”
of dimentions expressed by an exponent
is the minimum number of terms that are
necessary to express every term as a whole number:
Real Integer— Unit).
Noam Elkies was the Harvard professor
who back in 1988
had found a counterexample to Euler’s conjecture,
thereby proving that it was false:
(2,682,440)4 +
(15,365,639)4 +
(18,796,760)4 =
(20,615,673)4.
The solution for exponent
n = 4,
has only
3 terms
not 4 terms, as Euler and I
conned ourselves into the same jecture.
This clearly demonstrates:
my theory is not valid. Fortunately,
this was only
frivolity for me,
not the professional notoriety of Andrew Wiles.
Page 272 - 273,
Fermat’s
Enigma,
Simon Singh explains:
“ After one
or two days of turmoil some mathematicians
took a second look at the E-mail
and began to realize that,
although it was typically dated April 2
or April 3,
this was a result of having received it
second - or thirdhand.
The original message was dated April 1.
The E-mail was a mischievous hoax
perpetrated by the Canadian number theorist
Henri Darmon.”
Read all about it! This wonderful book
has the most clear, explicit explanations
of the triumph
of Andrew Wiles, and his
associates who reviewed his work.
Fraction terms are defined by Subtracting Exponents.
This Identity results from
n = zero:
Ax/Ax =
A(x - x) =
A0 ≡
1.
A0 ≠
B0 + C0 ≡
1 + 1 = 2.
There are no Integer solutions
for: n = zero.
[ The term n
is Non-Zero
in the theorem. ]
A0 = 1
is a special case of the concept of Unit.
The “Conceptual Common Denominator”
is a re-statement of the Law of Identity:
“Each thing is itself.”
The Identity does not specify any
Commensurable Relation (of extent)
with any “other thing.”
A ≡
A ,
The Axiom of Identity is a fundamental statement
of the nature of the universe: Each thing cannot be
what it is,
and
what it is not.
Inane Plato
infers (in an alley) that
Not any (Extent of a)
“thing”
which is “walking,” Exists in his empty room.
Plato alleges that the form of “walking”
occurs in that empty room.
“That archaic appraisal,” alleges Aristotle,
“is a crock O’diabolical allegation.
Walking is appurtenant to existing,
not zany ambulation of an apparition.
The activity of a “thing” is an effect
of what it is:
each “thing” has Identity.
Even egregious Zeno expostulates that
Zero “walking” Effect
is achieved by
Zero Extent
of existing Cause (“thing”).
This idealistic abstraction of conceptual comprehension
is myopic exacerbation of apocryphal,
apoplectic, misapprehension!”
Aristotle identifies the relation of
Identity—Being (Existing).
Each particular thing is a Unit
of particular Identity,
having a particular Extent in each Dimention of its existence.
Each Unit exists in a particular place
relative to any other things,
Existence is the Cause of any activity.
Identity of each Existent
limits the actions which it can,
and will perform, in relation with any other things
(Existents).
The Nature of each thing is the activity that it performs.
Identity of each existent explains the particular
Nature of that thing.
Science is the method of
non-contradictory Identification
of the Identity of things,
which achieve the activities
which are observed.
Repeated experiments confirm the relation
of Existence—Identity.
Existence is Identity.
Consciousness is Identification.
Atlas
Shrugged
(Ayn Rand) p. 942.
Exponent terms
generate exponential curves.
FERMAT’s equation
could apply to curved surfaces,
if the Solution were not
limited to integers.
My conclusion is:
At least n terms,
of specified extent in each Dimention, must be
added together to equate with
the term An.
The quantity of
Dimentions of the terms:
An,
Bn,
Cn;
is defined to be
the positive Integer
Exponent: n.
Mathematics is a language
of quantitative relations.
Introduction To
Objectivist Epistemology
by Ayn Rand.
The foundation of
my argument has been built
by Ayn Rand,
as summarized in the above quotes.
Definition of terms is the foundation of the Premise
that the expression
An is defined
by n terms,
which specify extent of a straight line (one Dimention)
in each of the
n Dimentions.
An ≡
An
is an identity when only one term is equated with
An.
But FERMAT equates the Sum of
two terms with
An.
If both terms must be
greater than zero, then the definition of a
specific extent for each Dimention is a consequence
of identity.
The pattern suggests that the quantity
of terms which are Summed
and equated with
An,
must be at least the same quantity as exponent
n.
The sum of an infinite series of
Non-Integer terms
can yield an integer.
The alternative
is the limitation of
Non-zero
Integer terms,
or the quantity of terms.
Each term in the equation
is defined by a
Real Integer Unit
of Extent
in a dimention.
The case of:
n = three (3)
A3 = B3
+ C3
Two Dementia brothers compete
in a drinking contest.
The Volume extent of each brother
in three Dimentions is:
Bourbon (in3) &
Champagne (in3).
A is Absinthe.
The two Dementia brothers will not maintain Integrity
when they drink Absinthe.
Deluded drunks with Imaginary multiple personalities
will hallucinate about the advertised Third Dementia
( n = 3)
of the drinking team.
Real Integer solutions disqualify any two ninny inebriates,
claiming to be three.
Timothy Leary’s Irrational brethren
may claim multiple Mental disability payments -
one for each imaginary alter-ego,
But they do not qualify for Fermat’s
calculating Realism and Integrity.
The exponent n
specifies the quantity of Dimentions which each term must have
as a commensurable attribute,
in order to be expressed in terms of a
Real Integer (unit).
When n = three,
The term: A3
specifies a cube.
Perhaps it is not a cube; but it is a construct of Mass,
Rotation, and elapsed Time.
One uses a hammer to drive a nail.
When a drunk uses a whiskey bottle for this purpose,
the principle is the same.
The “hammer” which does not move,
or the movement without elapsed time,
or the impact without the mass
which achieves force:
is similar to the term
A3,
with only two specified extents for Three Dimentions.
The third component can be imagined,
or derived from a vector of two dimentions.
Don’t suggest that Faith,
or Evolution will provide missing information.
When Exponent n is greater than two,
the two terms: Bn + Cn
are not sufficient to describe the extent
in Real Integer terms,
in each of the Identified quantity
of n dimentions.
Additional information must be
smuggled in -
by use of irrational numbers,
which are generated by extending from a common point,
in more than one Dimention.
Pi is an exchange ratio of
Curved Dimention,
relative to Straight Dimention.
The Premise is:
Real Integer solutions to
FERMAT’s Last Theorem
restrict each term to be defined in terms of straight
Dimentions which extend orthogonal:
not curved, and not vectors.
The Exponent of the term
An,
Identifies the Extent of only One
of n Dimentions.
The restriction of Non-Zero Integer solutions,
implies the minimum quantity of
Non-Zero Integer terms which can be Summed
to equate with:
An.
The Common Denominator must be a unit
which integrates through every multiple of the exponent power.
This is implied by the Commensurable Unit:
x, which is a
common factor of all terms.
A2 =
B2 + C2
(x * 5)2 =
(x * 3)2 +
(x * 4)2
and
A3 =
B3 + C3 +
D3
(x * 6)3 =
(x * 3)3 +
(x * 4)3 +
(x * 5)3
Every unit increase of the Exponent
n will increase
the gap generated by:
(A + 1)n - An.
The restriction of
Real Integer solutions
eliminates terms such as Pi,
and Imaginary numbers,
which are defined by more than one Dimention.
Was
Epimenides this Syllogismic ?
If the South had only wanted to
protect slavery . . ,
And did NOT ratify the
Original 13th Amendment,
Therefore
. . .
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